Elements of a space If A is a normed linear space and there exists an element x,y in A, then is it also true that x-y is also in A as well?
I am pretty sure this is true by definition, but I am not able to find any reference backing this, thereby leading myself to doubt the statement.
 A: Look at the definition for "normed linear space" in whatever text you're using. It will contain axioms saying:


*

*there is an addition operation sending any $x$ and $y$ in $A$ to $x+y$, also in $A$; and

*every element $x$ in $A$ has an additive inverse in $A$, which we denote $-x$, such that ...


or similar.
Therefore, if $x$ and $y$ are in $A$, we can define "$x-y$" to be shorthand notation for $x + (-y)$, which we know will be in $A$ from the above axioms. 
(Note that if your axioms don't give an "additive inverse" explicitly, they will at least allow you to perform scalar multiplication. We can then interpret "$-x$" as shorthand for the scalar multiplication $(-1)x$ and do the same trick.)
A: In fact, every linear combination $ax+by$ will be an element of the space for scalars $a,b$ and vectors $x,y$.
A: This is true even for ordinary linear spaces, e.g. Vector spaces. 
You can check out the axioms for vector space, it is closed under addition and the additive inverse exists, thus $x-y$ is in it.
